Graphs and Combinatorics

, Volume 3, Issue 1, pp 55–66 | Cite as

On the number of faces of centrally-symmetric simplicial polytopes

  • Richard P. Stanley


I. Bárány and L. Lovász [Acta Math. Acad. Sci. Hung.40, 323–329 (1982)] showed that ad-dimensional centrally-symmetric simplicial polytopeP has at least 2 d facets, and conjectured a lower bound for the numberf i ofi-dimensional faces ofP in terms ofd and the numberf0 =2n of vertices. Define integers\(h_0 ,...,h_d {\mathbf{ }}by{\mathbf{ }}\sum\limits_{i = 0}^d {f_{i - 1} } (x - 1)^{d - i} = \sum\limits_{i = 0}^d {h_i x^{d - i} } \) A. Björner conjectured (unpublished) that\(h_i \geqslant \left( {\begin{array}{*{20}c} d \\ i \\ \end{array} } \right)\) (which generalizes the result of Bárány-Lovász sincef d−1 =∑ h i ), and more strongly that\(h_i - h_{i - 1} \geqslant \left( {\begin{array}{*{20}c} d \\ i \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} d \\ {i - 1} \\ \end{array} } \right),1 \leqslant i \leqslant \left\lfloor {{d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2}} \right\rfloor \), which is easily seen to imply the conjecture of Bárány-Lovász. In this paper the conjectures of Björner are proved.


Acta Math Simplicial Polytopes 
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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Richard P. Stanley
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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